Question

Suppose $a,b,c$ are three non-zero real numbers. Then the equation $x^2+(a+b+c)x+a^2+b^2+c^2=0$ has

• A. rational and unequal roots.
• B. irrational roots.
• C. rational and equal roots.
• D. no real roots.

Answer 1

The correct option is D no real roots.

$x^2+(a+b+c)x+a^2+b^2+c^2=0$
$\Delta =(a+b+c{)}^2-4(a^2+b^2+c^2)$
$=-3a^2-3b^2-3c^2+2ab+2bc+2ca$
$=-[3a^2+3b^2+3c^2-2ab-2bc-2ca]$
$=-\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2+a^2+b^2+c^2]$
$\Rightarrow \Delta <0$
Hence, the roots are non-real.